* Prince William, Kate and baby still at hospital
* Family all doing well, officials say
* Grandfather Prince Charles is "thrilled"
The world could get its the first glimpse of Britain's new prince on Tuesday, with Prince William and his wife, Kate, expected to leave a London hospital with their baby son.
Kate, 31, gave birth to the couple's first child, who is third in line to the British throne, on Monday afternoon, ending weeks of feverish anticipation about the arrival.
Royal aides said the couple and their newborn were all doing well and expected to leave hospital after 1700 GMT on Tuesday, or possibly on Wednesday morning.
"We would like to thank the staff at the Lindo Wing and the whole hospital for the tremendous care the three of us have received," William and Kate said in a statement.
"We know it has been a very busy period for the hospital and we would like to thank everyone - staff, patients and visitors - for their understanding during this time."
The baby's name will be announced at a later date but George and James, both traditional royal names, were favoured choices with British bookmakers for the child, who is destined, one day, to be king.
The popular couple are expected to adhere to tradition by giving the public the first sight of the royal baby on the steps when they leave St Mary's Hospital in west London - just as William's father Prince Charles and late mother, Princess Diana, did with him.
"We're here to witness history, where a future monarch has been born. I just can't wait to see them today," said Maria Scott, a housewife from Newcastle in northern England who has camped outside the hospital since Saturday.
Kensington Palace announced the arrival of the boy, weighing 8 lb 6 oz (3.8 kg) at about 8:30 p.m., on Monday, four hours after his birth, saying Kate and her child were doing well and would remain in hospital overnight.
William, who said he and Kate could not be happier, was with them.
Their son is third in line to the throne after grandfather Charles and William, 31, and pushes the fun-loving Prince Harry, William's brother, into fourth place.
"I'm thrilled and very excited," said Charles during an official visit to northern England.
His second wife Camilla said: "I think mother, son and father are all well. And I think it's a wonderfully uplifting moment for the country," adding that Charles would make a "brilliant" grandfather.
Congratulations flooded in from around the world after the announcement of the birth, which was followed moment-by-moment by global media as well as TV and press in Britain.
British tabloid newspaper the Sun temporarily renamed itself the Son in honour of the baby, while the left-leaning Guardian newspaper provided readers of its website with a "Republican" button so that they could filter out the barrage of royal news if they wanted.
The birth fuels a new wave of popularity for the House of Windsor led by the younger royals, William and Harry, who were both born to Diana at St Mary's Hospital.
Support for the royals dipped after Diana's death in a car crash in Paris in 1997, a year after her divorce from Charles, as the royals were accused of being out of touch with modern Britain over their handling of the aftermath.
But last year's celebrations of Queen Elizabeth's 60th year on the throne showed support for the monarchy was running at a record high.
Hordes of TV crews and photographers, and royal fans wrapped in Union Jack flags, remained camped outside the hospital, waiting for the first photo of the baby, who will be called the Prince of Cambridge.
There will also be a 41-gun salute at London's Green Park and 62 rounds fired at the Tower of London to herald news of the birth.
Outside the Queen's Buckingham Palace London residence, a military band played "Congratulations" following the traditional Changing of the Guard ceremony.
William and Kate, who met when they were students at St Andrews University in Scotland about 10 years ago, have officially been known as the Duke and Duchess of Cambridge since their sumptuous royal wedding in April 2011.
The couple, who have been living in a cottage in north Wales where William is based as a Royal Air Force helicopter pilot, will eventually take up residence with their baby at Apartment 1A at London's Kensington Palace, William's childhood home, when a one million pound refurbishment is completed later this year.
Royal experts said after they leave the hospital, the new prince will be taken out of the public glare.
"All we will probably see is a glimpse of the top of the baby's head," said Joe Little, managing editor of Majesty magazine.
"After that we won't see them for some time. Having a baby is a very private moment and they are a private couple so the next time we see the baby will be the official photo and that could be weeks."
Not all Britons were celebrating the news however, with Britain's small Republican movement saying it was wrong the future head of state should be chosen by birth.
"Storms over London. God must be unhappy with media coverage too," the campaign group Republic tweeted as London was hit by a heavy downpour of rain.